
We first consider \((Y,{{G}})\)  is the minimal resolution of \((X,{{F}})\)  with exceptional divisors \(E_1\) , \(\ldots \) , \(E_r\) .
Let \(Z = \sum _{i=1}^ra_iE_i\)  where \(a_i\) 's are non-negative integers.
By the theorem on formal functions, it suffices to show that
\(h^1(Z,{\mathcal {O}}_Y(K_{{G}})\otimes {\mathcal {O}}_Z)=0\) 
for any effective (non-zero) divisor \(Z\) .
Let \(A := \sum _{i=1}^ra_i\) .
We will show the vanishing by induction on \(A\) .

When \(A = 1\) , there is exactly one \(i\)  such that \(a_i\)  is positive and equals to 1.
Without loss of generality, we assume \(a_1=1\)  and \(a_i=0\)  for \(i\ge 2\) .
Claim \(\deg (K_{E_1})-K_{{G}}\cdot E_1<0\) .

Proof

If \(E_1\)  is invariant, then we have \(K_{{G}}\cdot E_1\ge -1\)  and
\(\deg (K_{E_1})-K_{{G}}\cdot E_1 = -2-K_{{G}}\cdot E_1\le -1<0.\) 

If \(E_1\)  is non-invariant, then we have
\(\deg (K_{E_1})-K_{{G}}\cdot E_1 &=& 2p_a(E_1)-2+E_1^2 \\&\le & 2p_a(E_1)-2 - (2p_a(E_1)-1+\deg \,[E_1]) \\&=& -1-\deg \,[E_1] < 0.\) 


Thus, \(h^1(E_1,{\mathcal {O}}_Y(K_{{G}})\otimes {\mathcal {O}}_{E_1}) = h^0(E_1,\omega _{E_1}\otimes {\mathcal {O}}_{E_1}(-K_{{G}})) = 0\) .

Let \(Z_\ell = Z-E_\ell \)  for some \(\ell \)  with positive \(a_\ell \) , which will be determined later.
Then we combine the following two short exact sequences
\({0 [r] & {\mathcal {O}}_Y(-Z) [r] & {\mathcal {O}}_Y(-Z_\ell ) [r] & {\mathcal {O}}_{E_\ell }\otimes {\mathcal {O}}_{Y}(-Z_\ell ) [r] & 0}\) 
and
\({0 [r] & {\mathcal {O}}_Y(-Z_\ell )/{\mathcal {O}}_Y(-Z) [r] & {\mathcal {O}}_Z [r] & {\mathcal {O}}_{Z_\ell }[r] & 0}\) 
into a short exact sequecne
\({0 [r] & {\mathcal {O}}_{E_\ell }\otimes {\mathcal {O}}_{Y}(-Z_\ell ) [r] & {\mathcal {O}}_Z [r] & {\mathcal {O}}_{Z_\ell }[r] & 0}.\) 
Tensoring with \({\mathcal {O}}_Y(K_{{G}})\) , we get
\({0 [r] & {\mathcal {O}}_{E_\ell }\otimes {\mathcal {O}}_Y(K_{{G}}-Z_\ell ) [r] & {\mathcal {O}}_Y(K_{{G}})\otimes {\mathcal {O}}_Z [r] & {\mathcal {O}}_Y(K_{{G}})\otimes {\mathcal {O}}_{Z_\ell }[r] & 0}.\) 
Now, by induction hypothesis, we have \(h^1(Z_\ell ,{\mathcal {O}}_Y(K_{{G}})\otimes {\mathcal {O}}_{Z_\ell })=0\) .
Therefore, it suffices to show \(h^1(E_\ell ,{\mathcal {O}}_{E_\ell }\otimes {\mathcal {O}}_Y(K_{{G}}-Z_\ell ))=0\) .
This follows from
\(K_{{G}}\cdot E_\ell -Z_i\cdot E_\ell >\deg (K_{E_\ell }) = K_Y\cdot E_\ell +E_\ell ^2,\) 
which is equivalent to
\((K_{{G}}-K_Y-Z)\cdot E_\ell >0.\) 

Let \(K_{{G}}-K_Y\equiv _f\sum _{i=1}^rb(E_i)E_i\)  for some \(b(E_i)\in {\mathbb {R}}\) .
We will denote \(b(E_i)\)  by \(b_i\)  as well.
Claim \(b(E_i)\le 1\)  for all \(i\) .
Moreover, either \(b(E_i)=1\)  for all \(i\)  or \(b(E_i)<1\)  for all \(i\) .

Proof

If \(E_j\)  is invariant, then we have
\((K_{{G}}-K_Y)\cdot E_j &=& \textnormal {Z}({{G}},E_j)-2+E_j^2+2 \\&=&\textnormal {Z}({{G}},E_j)+E_j^2 \ge \left(\sum _{i=1}^rE_i\right)\cdot E_j.\) 

If \(E_j\)  is non-invariant, then we have
\((K_{{G}}-K_Y)\cdot E_j &=& -E_j^2 + E_j^2+2-2p_a(E_j) \\&> & \deg \,[E_j]+E_j^2 = \left(\sum _{i=1}^rE_i\right)\cdot E_j.\) 

By Lemma REF , we have that \(b_i\le 1\)  for all \(i\) .
Moreover, \(\bigcup _{i=1}^rE_i\)  is connected, we have either \(b_i=1\)  for all \(i\)  or \(b_i<1\)  for all \(i\) .

Claim We have \(b(E_i)\ge 0\)  for all \(i\)  except for the non-invariant \(E_0\)  of (6) and (7) in Theorem REF .

Proof Let \(2D\)  be the sum of all \((-1)\) -\({{G}}\) -curves whose self-intersections are \(-2\) .
Then we have \((K_{{G}}-K_Y)\cdot E_i\le D\cdot E_i\)  for all \(i\) .
Thus, by Lemma REF , we have that \(b_i\ge 0\)  for all \(i\) .
In the case of type (6) and (7), we note that
\((K_{{G}}-K_Y)\cdot E_0\le 2-2p_a(E_0) \le -E_0^2\) 
and
\((K_{{G}}-K_Y)\cdot E_j = 1+E_j^2\le -1 = -E_0\cdot E_j\) 
for any invariant curve \(E_j\) .
Then, by Lemma REF , we have \(b(E_j)\ge 0\)  for all invariant curves \(E_j\) .



Let \(W = \sum _{i=1}^rc_iE_i\)  be an \(f\) -anti-ample effective divisor where \(c_i>0\)  for all \(i\) .
We have following two cases:

If \(a_i-b_i\)  is positive for some \(i\) , then we define \(\alpha \in {\mathbb {R}}_{\ge 0}\)  to be the maximal number such that \(\alpha (a_i-b_i)\le c_i\)  for all \(i\) .
Then there is in index \(\ell \)  such that \(c_\ell -\alpha (a_\ell -b_\ell )=0\) , and therefore
\(\big (W+\alpha (K_{{G}}-K_Y-Z)\big )\cdot E_\ell = \left(\sum _{i=1}^r \big (c_i-\alpha (a_i-b_i)\big )E_i\right)\cdot E_\ell \ge 0.\) 
Hence, we have
\((K_{{G}}-K_Y-Z)\cdot E_\ell \ge \frac{-1}{\alpha }W\cdot E_\ell >0.\) 

If \(a_i-b_i\le 0\)  for all \(i\) , then \(a_i\le b_i\le 1\)  for all \(i\) .
Thus, \(b_i=1\)  for all \(i\) , otherwise, we have \(b_i<1\)  for all \(i\) , and then \(a_i=0\)  for all \(i\)  since \(a_i\) 's are non-negative integers.


Suppose \(\bigcup _{i=1}^rE_i\)  forms a dual graph for a canonical foliation singularity.
Then we have \(b(E_i)\ge 0\)  for all \(i\) .
If \(b_i<1\)  for all \(i\) , then, from \(c_\ell -\alpha (a_\ell -b_\ell )=0\) , we have \(a_\ell = b_\ell +\frac{c_\ell }{\alpha }>0\) .
If \(b_i=1\)  for all \(i\) , then \(a_i\le 1\)  for all \(i\)  and \(\bigcup _{i=1}^rE_i\)  is an elliptic Gorenstein leaf \(\Gamma \) .
By [1]}, we have \({\mathcal {O}}_\Gamma (K_{{G}})\)  is not torsion and has degree 0.
Also \(\Gamma \)  is Cohen-Macaulay with trivial dualizing sheaf, then by Serre duality, we have
\(h^1(\Gamma ,{\mathcal {O}}_Y(K_{{G}})\otimes {\mathcal {O}}_\Gamma ) = h^0(\Gamma ,{\mathcal {O}}_Y(-K_{{G}})\otimes {\mathcal {O}}_\Gamma ) = 0.\) 
Thus, we have shown the case when \(a_i=1\)  for all \(i\) .
Therefore, when \(a_i=0\)  for some \(i\) , we have an index \(j\)  such that
\((K_{{G}}-K_Y-Z)\cdot E_j = \left(\sum _i(b_i-a_i)E_i\right)\cdot E_j> 0.\) 

Now suppose \(\bigcup _{i=1}^rE_i\)  forms a dual graph for a non-canonical foliation singularity.
Note that we have \(b_i<1\)  for all \(i\) .
If \(E_\ell \)  in step (REF ) is invariant, then from \(c_\ell -\alpha (a_\ell -b_\ell )=0\) , we have \(a_\ell = b_\ell +\frac{c_\ell }{\alpha }>0\) .
Thus, it suffices to consider the case when \(E_\ell \)  is non-invariant and \(a_\ell = 0\) .
Then the support of \(Z = \sum _iZ_i = \sum _ia_iE_i\)  is the disjoint union of \((-1)\) -\({{G}}\) -curves \(E_i\)  where \(Z_i = a_iE_i\) .
Note that
\(H^1(Z,{\mathcal {O}}_Y(K_{{G}})\otimes {\mathcal {O}}_Z)=\bigoplus _i H^1(Z_i,{\mathcal {O}}_Y(K_{{G}})\otimes {\mathcal {O}}_{Z_i}).\) 
Since the contraction of \(E_i\)  introduces a terminal foliation singularity, we have \(h^1(Z_i,{\mathcal {O}}_Y(K_{{G}})\otimes {\mathcal {O}}_{Z_i}) = 0\)  for all \(i\) , and thus \(h^1(Z,{\mathcal {O}}_Y(K_{{G}})\otimes {\mathcal {O}}_Z)=0\) .

Now we consider the general resolution.
Note that any general resolution \(f : (Y,{{G}}) \rightarrow (X,{{F}})\)  factors through the minimal resolution \(h : (Z,{{H}}) \rightarrow (X,{{F}})\) .
Let \(f = h\circ g\) .
So \(g\)  is the composition of blowups.
We have shown that \(R^1h_*{\mathcal {O}}_Z(K_{{H}}) = 0\) .
Notice that \(g_*{\mathcal {O}}_Y(K_{{G}}) = {\mathcal {O}}_Z(K_{{H}})\) .
By Grothendieck spectral sequence, we have the following exact sequence:
\({0 [r] & R^1h_*(g_*{\mathcal {O}}_Y(K_{{G}})) [r] & R^1f_*{\mathcal {O}}_Y(K_{{G}}) [r] & h_*R^1g_*{\mathcal {O}}_Y(K_{{G}}).}\) 
Since \(R^1h_*(g_*{\mathcal {O}}_Y(K_{{G}})) = R^1h_*{\mathcal {O}}_Z(K_{{H}})=0\)  and \(R^1g_*{\mathcal {O}}_Y(K_{{G}})=0\)  by Lemma REF , we have \(R^1f_*{\mathcal {O}}_Y(K_{{G}})=0\) .
\(\Box \)


